George Bush's personal interest in Mongolia might be thought limited. Yet, when the country's then leader visited Washington last year, the US president enthusiastically declared "a new era of comprehensive partnership".

Mongolia's 2.6 million people occupy an area of 604,000 square miles (the UK has nearly 60 million people in 95,000 square miles). While Mongolia has oil, its main resource is 20 million sheep and goats. But ruminants were not the reason Mr Bush was all riled up.

Mongolia is geographically sandwiched between China and Russia. And it has been steadily drawn into what Walter Russell Mead of the Council on Foreign Relations in New York calls the "strategic net" being woven by the US in Asia to "persuade China to keep its ambitions within reason".

Alarmed by China's rapid rise, and to a lesser extent by its developing collaboration with Russia, the US is pushing back. Results range from a sudden warming of relations with Vietnam to plans for "strategic partnership" with India.

"China has become one of the largest traders and investors with many Asian countries," Christopher Hill, assistant secretary of state for east Asia, told the Senate in June. But he noted that while US investment in south-east Asian countries totalled over $85bn (about£48bn), China's amounted to only $2bn.

While Beijing wielded influence in places such as North Korea, Burma and Cambodia, Mr Hill pointed to America's "strong and enduring alliances with Japan, Australia, South Korea, Thailand and the Philippines". In terms of global power politics, he suggested, it was no contest.

The corollary to stepped-up US efforts to contain and integrate China is steady American pressure on Russia via Moscow's former Soviet satellites.

Even as US forces were ordered out of Uzbekistan, which has agreed energy deals with Moscow and Beijing, leases on US military bases in Kyrgyzstan and Tajikistan were renewed last month.

And despite Vladimir Putin's warnings to foreign-funded NGOs not to dabble in politics in the former Soviet republics, Washington shows no sign of backing off.

To the west, the US and its favoured ally, Poland, seem to be preparing another pro-democracy front in Belarus, where Alexander Lukashenko's authoritarian regime faces elections next year.

And in Georgia and Ukraine, scene of the original western-backed "colour revolutions" that the Kremlin fears could be emulated at home, the US is being urged to do more.

"We have a political understanding that something should be done and there can be some kind of US role," he told the Wall Street Journal Europe.

America undoubtedly faces a long-term battle for influence in the Asia-Pacific area. But, says Fiona Hill of the Brookings Institution in Washington, there are present-day limits to what even Beijing and Moscow combined can do in response to the US's "soft power" offensives. "China has got a lot of internal problems such as unrest in the rural areas and energy is its Achilles heel," Ms Hill said. To keep growing, China needed to keep vital export markets open, principally in the US. For this and other reasons, for example, experts say direct confrontation over Taiwan is unlikely.

"Russia gets infuriated with the US but at the same time it is incredibly worried about China. Some [Russian] military analysts say, 'China could turn against us but we are selling them all these arms. Is this sensible?'" Ms Hill said.

The upshot, according to a UK Defence Academy study by Mark Smith, is that while pursuing closer ties, "neither side has any interest in creating a formal anti-American alliance".

While China may ultimately go it alone, "fear of being marginalised has played a large part in driving the Putin leadership towards strategic partnership with the US and the EU," Mr Smith said.

In this sense, closer China-Russia collaboration reflects current weakness, not strength. As the US spins its web, expect more Mongolian-style mutton diplomacy.